Editing Propositional calculus (section)

==== Semantics via assignment expressions ====
Some authors (viz., all the authors cited in this subsection) write out the connective semantics using a list of statements instead of a table. In this format, where <math>\mathcal{I}(\varphi)</math> is the interpretation of <math>\varphi</math>, the five connectives are defined as:<ref name="BostockIntermediate" /><ref name=":29" />

* <math>\mathcal{I}(\neg P) = \mathsf{T}</math> if, and only if, <math>\mathcal{I}(P) = \mathsf{F}</math>
* <math>\mathcal{I}(P \land Q) = \mathsf{T}</math> if, and only if, <math>\mathcal{I}(P) = \mathsf{T}</math> and <math>\mathcal{I}(Q) = \mathsf{T}</math>
* <math>\mathcal{I}(P \lor Q) = \mathsf{T}</math> if, and only if, <math>\mathcal{I}(P) = \mathsf{T}</math> or <math>\mathcal{I}(Q) = \mathsf{T}</math>
* <math>\mathcal{I}(P \to Q) = \mathsf{T}</math> if, and only if, it is true that, if <math>\mathcal{I}(P) = \mathsf{T}</math>, then <math>\mathcal{I}(Q) = \mathsf{T}</math>
* <math>\mathcal{I}(P \leftrightarrow Q) = \mathsf{T}</math> if, and only if, it is true that <math>\mathcal{I}(P) = \mathsf{T}</math> if, and only if, <math>\mathcal{I}(Q) = \mathsf{T}</math>
Instead of <math>\mathcal{I}(\varphi)</math>, the interpretation of <math>\varphi</math> may be written out as <math>|\varphi|</math>,<ref name="BostockIntermediate" /><ref name="ms28"/> or, for definitions such as the above, <math>\mathcal{I}(\varphi) = \mathsf{T}</math> may be written simply as the English sentence "<math>\varphi</math> is given the value <math>\mathsf{T}</math>".<ref name=":29" /> Yet other authors<ref name="ms29"/><ref name=":43"/> may prefer to speak of a [[Model theory|Tarskian model]] <math>\mathfrak{M}</math> for the language, so that instead they'll use the notation <math>\mathfrak{M} \models \varphi</math>, which is equivalent to saying <math>\mathcal{I}(\varphi) = \mathsf{T}</math>, where <math>\mathcal{I}</math> is the interpretation function for <math>\mathfrak{M}</math>.<ref name=":43" />